17. JAX

This lecture provides a short introduction to Google JAX.

JAX is a high-performance scientific computing library that provides

• a NumPy-like interface that can automatically parallelize across CPUs and GPUs,

• a just-in-time compiler for accelerating a large range of numerical operations, and

• automatic differentiation.

Increasingly, JAX also maintains and provides more specialized scientific computing routines, such as those originally found in SciPy.

In addition to what’s in Anaconda, this lecture will need the following libraries:

We’ll use the following imports

17.1JAX as a NumPy Replacement¶

Let’s look at the similarities and differences between JAX and NumPy.

17.1.1Similarities¶

Above we import jax.numpy as jnp, which provides a NumPy-like interface to array operations.

One of the attractive features of JAX is that, whenever possible, this interface conform to the NumPy API.

As a result, we can often use JAX as a drop-in NumPy replacement.

Here are some standard array operations using jnp:

It should be remembered, however, that the array object a is not a NumPy array:

Even scalar-valued maps on arrays return JAX arrays rather than scalars!

17.1.2Differences¶

Let’s now look at some differences between JAX and NumPy array operations.

17.1.2.1Speed!¶

One major difference is that JAX is faster --- and sometimes much faster.

To illustrate, suppose that we want to evaluate the cosine function at many points.

17.1.2.1.1With NumPy¶

Let’s try with NumPy

And one more time.

Here

• NumPy uses a pre-built binary for applying cosine to an array of floats

• The binary runs on the local machine’s CPU

17.1.2.1.2With JAX¶

Now let’s try with JAX.

Let’s time the same procedure.

Now let’s time it again.

On a GPU, this code runs much faster than its NumPy equivalent.

Also, typically, the second run is faster than the first due to JIT compilation.

This is because even built in functions like jnp.cos are JIT-compiled --- and the first run includes compile time.

Why would JAX want to JIT-compile built in functions like jnp.cos instead of just providing pre-compiled versions, like NumPy?

The reason is that the JIT compiler wants to specialize on the size of the array being used (as well as the data type).

The size matters for generating optimized code because efficient parallelization requires matching the size of the task to the available hardware.

17.1.2.2Size Experiment¶

We can verify the claim that JAX specializes on array size by changing the input size and watching the runtimes.

The run time increases and then falls again (this will be more obvious on the GPU).

This is in line with the discussion above -- the first run after changing array size shows compilation overhead.

Further discussion of JIT compilation is provided below.

17.1.2.3Precision¶

Another difference between NumPy and JAX is that JAX uses 32 bit floats by default.

This is because JAX is often used for GPU computing, and most GPU computations use 32 bit floats.

Using 32 bit floats can lead to significant speed gains with small loss of precision.

However, for some calculations precision matters.

In these cases 64 bit floats can be enforced via the command

Let’s check this works:

17.1.2.4Immutability¶

As a NumPy replacement, a more significant difference is that arrays are treated as immutable.

For example, with NumPy we can write

and then mutate the data in memory:

In JAX this fails 😱.

The designers of JAX chose to make arrays immutable because

1. JAX uses a functional programming style and

2. functional programming typically avoids mutable data

We discuss these ideas below.

17.1.2.5A Workaround¶

JAX does provide a direct alternative to in-place array modification via the at method.

Applying at[0].set(1) returns a new copy of a with the first element set to 1

Obviously, there are downsides to using at:

• The syntax is cumbersome and

• we want to avoid creating fresh arrays in memory every time we change a single value!

Hence, for the most part, we try to avoid this syntax.

(Although it can in fact be efficient inside JIT-compiled functions -- but let’s put this aside for now.)

17.2Functional Programming¶

From JAX’s documentation:

When walking about the countryside of Italy, the people will not hesitate to tell you that JAX has “una anima di pura programmazione funzionale”.

In other words, JAX assumes a functional programming style.

17.2.1Pure functions¶

The major implication is that JAX functions should be pure.

Pure functions have the following characteristics:

1. Deterministic

2. No side effects

Deterministic means

• Same input $\implies$ same output

• Outputs do not depend on global state

In particular, pure functions will always return the same result if invoked with the same inputs.

No side effects means that the function

• Won’t change global state

• Won’t modify data passed to the function (immutable data)

17.2.2Examples -- Pure and Impure¶

Here’s an example of a impure function

This function fails to be pure because

• side effects --- it modifies the global variable prices

• non-deterministic --- a change to the global variable tax_rate will modify function outputs, even with the same input array prices.

Here’s a pure version

This is pure because

• all dependencies explicit through function arguments

• and doesn’t modify any external state

17.2.3Why Functional Programming?¶

At QuantEcon we love pure functions because they

• Help testing: each function can operate in isolation

• Promote deterministic behavior and hence reproducibility

• Prevent bugs that arise from mutating shared state

The JAX compiler loves pure functions and functional programming because

• Data dependencies are explicit, which helps with optimizing complex computations

• Pure functions are easier to differentiate (autodiff)

• Pure functions are easier to parallelize and optimize (don’t depend on shared mutable state)

Another way to think of this is as follows:

JAX represents functions as computational graphs, which are then compiled or transformed (e.g., differentiated)

These computational graphs describe how a given set of inputs is transformed into an output.

JAX’s computational graphs are pure by construction.

JAX uses a functional programming style so that user-built functions map directly into the graph-theoretic representations supported by JAX.

17.3Random numbers¶

Random number generation in JAX differs significantly from the patterns found in NumPy or MATLAB.

17.3.1NumPy / MATLAB Approach¶

In NumPy / MATLAB, generation works by maintaining hidden global state.

Each time we call a random function, the hidden state is updated:

This function is not pure because:

• It’s non-deterministic: same inputs, different outputs

• It has side effects: it modifies the global random number generator state

This is dangerous under parallelization --- must carefully control what happens in each thread.

17.3.2JAX¶

In JAX, the state of the random number generator is controlled explicitly.

First we produce a key, which seeds the random number generator.

Now we can use the key to generate some random numbers:

If we use the same key again, we initialize at the same seed, so the random numbers are the same:

To produce a (quasi-) independent draw, one option is to “split” the existing key:

The following diagram illustrates how split produces a tree of keys from a single root, with each key generating independent random draws.

This syntax will seem unusual for a NumPy or Matlab user --- but will make more sense when we get to parallel programming.

The function below produces k (quasi-) independent random n x n matrices using split.

This function is pure

• Deterministic: same inputs, same output

• No side effects: no hidden state is modified

17.3.3Benefits¶

As mentioned above, this explicitness is valuable:

• Reproducibility: Easy to reproduce results by reusing keys

• Parallelization: Control what happens on separate threads

• Debugging: No hidden state makes code easier to test

• JIT compatibility: The compiler can optimize pure functions more aggressively

17.4JIT Compilation¶

The JAX just-in-time (JIT) compiler accelerates execution by generating efficient machine code that varies with both task size and hardware.

We saw the power of JAX’s JIT compiler combined with parallel hardware when we above, when we applied cos to a large array.

Here we study JIT compilation for more complex functions

17.4.1With NumPy¶

We’ll try first with NumPy, using

Let’s run with large x

Eager execution model

• Each operation is executed immediately as it is encountered, materializing its result before the next operation begins.

Disadvantages

• Minimal parallelization

• Heavy memory footprint --- produces many intermediate arrays

• Lots of memory read/write

17.4.2With JAX¶

As a first pass, we replace np with jnp throughout:

Now let’s time it.

The outcome is similar to the cos example --- JAX is faster, especially on the second run after JIT compilation.

This is because the individual array operations are parallelized on the GPU

But we are still using eager execution

• lots of memory due to intermediate arrays

• lots of memory read/writes

Also, many separate kernels launched on the GPU

17.4.3Compiling the Whole Function¶

Fortunately, with JAX, we have another trick up our sleeve --- we can JIT-compile the entire function, not just individual operations.

The compiler fuses all array operations into a single optimized kernel

Let’s try this with the function f:

The runtime has improved again --- now because we fused all the operations

• Aggressive optimization based on entire computational sequence

• Eliminates multiple calls to the hardware accelerator

The memory footprint is also much lower --- no creation of intermediate arrays

Incidentally, a more common syntax when targeting a function for the JIT compiler is

17.4.4How JIT compilation works¶

When we apply jax.jit to a function, JAX traces it: instead of executing the operations immediately, it records the sequence of operations as a computational graph and hands that graph to the XLA compiler.

XLA then fuses and optimizes the operations into a single compiled kernel tailored to the available hardware (CPU, GPU, or TPU).

The first call to a JIT-compiled function incurs compilation overhead, but subsequent calls with the same input shapes and types reuse the cached compiled code and run at full speed.

17.4.5Compiling non-pure functions¶

While JAX will not usually throw errors when compiling impure functions, execution becomes unpredictable!

Here’s an illustration of this fact:

In the code above, the global value a=1 is fused into the jitted function.

Even if we change a, the output of f will not be affected --- as long as the same compiled version is called.

Changing the dimension of the input triggers a fresh compilation of the function, at which time the change in the value of a takes effect:

Moral of the story: write pure functions when using JAX!

17.5Vectorization with vmap¶

Another powerful JAX transformation is jax.vmap, which automatically vectorizes a function written for a single input so that it operates over batches.

This avoids the need to manually write vectorized code or use explicit loops.

17.5.1A simple example¶

Suppose we have a function that computes the difference between mean and median for an array of numbers.

We can apply it to a single vector:

Now suppose we have a matrix and want to compute these statistics for each row.

Without vmap, we’d need an explicit loop:

However, Python loops are slow and cannot be efficiently compiled or parallelized by JAX.

With vmap, we can avoid loops and keep the computation on the accelerator:

17.5.2Combining transformations¶

One of JAX’s strengths is that transformations compose naturally.

For example, we can JIT-compile a vectorized function:

This composition of jit, vmap, and (as we’ll see next) grad is central to JAX’s design and makes it especially powerful for scientific computing and machine learning.

17.6Automatic differentiation: a preview¶

JAX can use automatic differentiation to compute gradients.

This can be extremely useful for optimization and solving nonlinear systems.

Here’s a simple illustration involving the function $f(x) = x^2 / 2$:

Let’s plot the function and derivative, noting that $f'(x) = x$.

Automatic differentiation is a deep topic with many applications in economics and finance. We provide a more thorough treatment in .

17.7Exercises¶

M = 10_000_000

n, β, K = 20, 0.99, 100
μ, ρ, ν, S0, h0 = 0.0001, 0.1, 0.001, 10, 0

@jax.jit
def compute_call_price_jax(β=β,
                           μ=μ,
                           S0=S0,
                           h0=h0,
                           K=K,
                           n=n,
                           ρ=ρ,
                           ν=ν,
                           M=M,
                           key=jax.random.key(1)):

    s = jnp.full(M, np.log(S0))
    h = jnp.full(M, h0)

    def update(i, loop_state):
        s, h, key = loop_state
        key, subkey = jax.random.split(key)
        Z = jax.random.normal(subkey, (2, M))
        s = s + μ + jnp.exp(h) * Z[0, :]
        h = ρ * h + ν * Z[1, :]
        new_loop_state = s, h, key
        return new_loop_state

    initial_loop_state = s, h, key
    final_loop_state = jax.lax.fori_loop(0, n, update, initial_loop_state)
    s, h, key = final_loop_state

    expectation = jnp.mean(jnp.maximum(jnp.exp(s) - K, 0))

    return β**n * expectation

with qe.Timer():
    compute_call_price_jax().block_until_ready()

with qe.Timer():
    compute_call_price_jax().block_until_ready()